Weighted Solyanik estimates for the strong maximal function

Abstract

Let ${\mathsf{M}}_{\mathsf{S}}$ denote the strong maximal operator on ${\mathbb{R}}^n$ and let $w$ be a non-negative, locally integrable function. For $\alpha \in (0,1)$ we define the weighted sharp Tauberian constant ${\mathsf{C}}_{\mathsf{S}}$ associated with ${\mathsf{M}}_{\mathsf{S}}$ by $${\mathsf{C}}_{\mathsf{S}}(\alpha ):=\underset{\begin{array}{c}E\subset {\mathbb{R}}^n\\ 0<w(E)<+\infty \end{array}}{\sup }\frac{1}{w(E)}w(\{x\in {\mathbb{R}}^n:{\mathsf{M}}_{\mathsf{S}}(1_E)(x)>\alpha \}).$$ We show that ${\lim }_{\alpha \to 1^{-}}{\mathsf{C}}_{\mathsf{S}}(\alpha )=1$ if and only if $w\in A_{\infty }^{\ast }$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate ${\mathsf{C}}_{\mathsf{S}}(\alpha )-1{\lesssim }_n(1-\alpha {)}^{(cn[w{]}_{A_{\infty }^{\ast }}{)}^{-1}}$ as $\alpha \to 1^{-}$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w{]}_{A_{\infty }^{\ast }})$ can not be improved in terms of $[w{]}_{A_{\infty }^{\ast }}$. As corollaries, we obtain a sharp reverse Hölder inequality for strong Muckenhoupt weights in ${\mathbb{R}}^n$ as well as a quantitative imbedding of $A_{\infty }^{\ast }$ into $A_p^{\ast }$. We also consider the strong maximal operator on ${\mathbb{R}}^n$ associated with the weight $w$ and denoted by ${\mathsf{M}}_{\mathsf{S}}^w$. In this case the corresponding sharp Tauberian constant ${\mathsf{C}}_{\mathsf{S}}^w$ is defined by $${\mathsf{C}}_{\mathsf{S}}^w\alpha ):=\underset{\begin{array}{c}E\subset {\mathbb{R}}^n\\ 0<w(E)<+\infty \end{array}}{\sup }\frac{1}{w(E)}w(\{x\in {\mathbb{R}}^n:{\mathsf{M}}_{\mathsf{S}}^w(1_E)(x)>\alpha \}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that ${\mathsf{C}}_{\mathsf{S}}^w(\alpha )-1{\lesssim }_{w,n}(1-\alpha {)}^{c_{w,n}}$ as $\alpha \to 1^{-}$ whenever $w\in A_{\infty }^{\ast }$ is a strong Muckenhoupt weight.

19 pages, submitted for publication

References in corpus (4)

• Sharp Reverse Hölder property for A_\infty weights on spaces of homogeneous type
• Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
• Variations on the Theme of Journe's Lemma
• Weighted Solyanik Estimates for the Hardy-Littlewood maximal operator and embedding of $A_{\infty }$ into $A_p$

Cited by in corpus (6)

• Sparse Domination for Bi-Parameter Operators Using Square Functions
• On the maximal directional Hilbert transform in three dimensions
• Asymptotically sharp reverse Hölder inequalities for flat Muckenhoupt weights
• On the Boundedness of Multilinear Fractional Strong Maximal Operator with multiple weights
• Reverse Hölder Property for strong weights and general measures
• A note on local Hölder continuity of weighted Tauberian functions